Constraining Lorentz-violating, modified dispersion
relations with gravitational waves
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Mirshekari, Saeed, Nicolás Yunes, and Clifford Will.
“Constraining Lorentz-violating, Modified Dispersion Relations
with Gravitational Waves.” Physical Review D 85.2 (2012): Web.
11 May 2012. © 2012 American Physical Society
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Detailed Terms
PHYSICAL REVIEW D 85, 024041 (2012)
Constraining Lorentz-violating, modified dispersion relations with gravitational waves
Saeed Mirshekari,1 Nicolás Yunes,2,3 and Clifford M. Will1
1
McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis Missouri 63130 USA
2
MIT and Kavli Institute, Cambridge, Massachusetts 02139, USA
3
Department of Physics, Montana State University, Bozeman, Montana 59717, USA
(Received 12 October 2011; published 25 January 2012)
Modified gravity theories generically predict a violation of Lorentz invariance, which may lead to a
modified dispersion relation for propagating modes of gravitational waves. We construct a parametrized
dispersion relation that can reproduce a range of known Lorentz-violating predictions and investigate their
impact on the propagation of gravitational waves. A modified dispersion relation forces different
wavelengths of the gravitational-wave train to travel at slightly different velocities, leading to a modified
phase evolution observed at a gravitational-wave detector. We show how such corrections map to the
waveform observable and to the parametrized post-Einsteinian framework, proposed to model a range of
deviations from General Relativity. Given a gravitational-wave detection, the lack of evidence for such
corrections could then be used to place a constraint on Lorentz violation. The constraints we obtain are
tightest for dispersion relations that scale with small power of the graviton’s momentum and deteriorate
for a steeper scaling.
DOI: 10.1103/PhysRevD.85.024041
PACS numbers: 04.30.w, 04.30.Nk, 04.50.Kd
I. INTRODUCTION
After a century of experimental success, Einstein’s fundamental theories, i.e. the special theory of relativity and
the General theory of Relativity (GR), are beginning to be
questioned. As an example, consider the observation of
ultra-high-energy cosmic rays. In relativity, there is a
threshold of 5 1019 eV (GZK limit) for the amount
of energy that charged particles can carry, while cosmic
rays have been detected with higher energies [1]. On the
theoretical front, theories of quantum gravity also generically predict a deviation from Einstein’s theory at sufficiently large energies or small scales. In particular, Lorentz
violation seems ubiquitous in such theories. These considerations motivate us to study the effects of Lorentz violation on gravitational-wave observables.
Einstein’s theory will soon be put to the test through a
new type of observation: gravitational waves (GWs). Such
waves are (far-field) oscillations of spacetime that encode
invaluable and detailed information about the source that
produced them. For example, the inspiral, merger and
ringdown of compact objects (black holes or neutron stars)
are expected to produce detectable waves that will access
horizon-scale curvatures and energies. Gravitational waves
may thus provide new hints as to whether Einstein’s theory
remains valid in this previously untested regime.
Gravitational-wave detectors are today a reality.
Ground-based interferometers, such as the Advanced
Laser Interferometer Gravitational Observatory (Ad.
LIGO) [2–4] and Advanced Virgo [5], are currently being
updated, and are scheduled to begin data acquisition by
2015. Second-generation detectors, such as the Einstein
Telescope (ET) [6,7] and the Laser Interferometer Space
Antenna (LISA) [8,9], are also being planned for the next
1550-7998= 2012=85(2)=024041(12)
decade. Recent budgetary constraints in the United States
have cast doubt on the status of LISA, but the European
Space Agency is still considering a descoped, LISA-like
mission (an NGO, or New Gravitational Observatory). The
detection of gravitational waves is, of course, not a certainty, as the astrophysical event rate is highly uncertain.
However, there is consensus that advanced ground detectors should observe a few gravitational-wave events by the
end of this decade.
Some alternative gravity theories endow the graviton
with a mass [10]. Massive gravitons would travel slower
than the speed of light, but most importantly, their speed
would depend on their energy or wavelength. Since gravitational waves emitted by compact binary inspirals chirp in
frequency, gravitons emitted in the early inspiral will travel
more slowly than those emitted close to merger, leading to
a frequency-dependent gravitational-wave dephasing,
compared to the phasing of a massless general relativistic
graviton. If such a dephasing is not observed, then one
could place a constraint on the graviton mass [11]. A
Lorentz-violating graviton dispersion relation leaves an
additional imprint on the propagation of gravitational
waves, irrespective of the generation mechanism. Thus a
bound on the dephasing effect could also bound the degree
of Lorentz violation.
In this paper, we construct a framework to study the
impact of a Lorentz-violating dispersion relation on the
propagation of gravitational waves. We begin by proposing
a generic, but quantum-gravitational inspired, modified
dispersion relation, given by
E2 ¼ p2 c2 þ m2g c4 þ Ap c ;
(1)
where mg is the mass of the graviton and A and are
two Lorentz-violating parameters that characterize the GR
024041-1
Ó 2012 American Physical Society
MIRSHEKARI, YUNES, AND WILL
PHYSICAL REVIEW D 85, 024041 (2012)
deviation ( is dimensionless while A has dimensions of
½energy2 ). We will assume that A=ðcpÞ2 1. When
either A ¼ 0 or ¼ 0, the modification reduces to that of a
massive graviton. When ¼ ð3; 4Þ, one recovers predictions of certain quantum-gravitation inspired models. This
modified dispersion relation introduces Lorentz-violating
deviations in a continuous way, such that when the parameter A is taken to zero, the dispersion relation reduces
to that of a simple massive graviton.
The dispersion relation of Eq. (1) modifies the gravitational waveform observed at a detector by correcting the
phase with certain frequency-dependent terms. In the
stationary-phase approximation (SPA), the Fourier transform of the waveform is corrected by a term of the form
ðAÞu1 , where u ¼ Mf is a dimensionless measure
of the gravitational-wave frequency with M the so-called
‘‘chirp mass.’’ We show that such a modification can be
easily mapped to the recently proposed parametrized postEinsteinian (ppE) framework [12,13] for an appropriate
choice of ppE parameters.
In deriving the gravitational-wave Fourier transform we
must assume a functional form for the waveform as emitted
at the source so as to relate the time of arrival at the
detector to the gravitational-wave frequency. In principle,
this would require a prediction for the equations of motion
and gravitational-wave emission for each Lorentzviolating theory under study. However, few such theories
have reached a sufficient state of development to produce
such predictions. On the other hand, it is reasonable to
assume that the predictions will be not too different from
those of general relativity. For example, we argued [11]
that for a theory with a massive graviton, the differences
would be of order ð=g Þ2 , where is the gravitational
wavelength, and g is the graviton Compton wavelength,
and g for sources of interest. Similar behavior might
be expected in Lorentz-violating theories. The important
phenomenon is the accumulation of dephasing over the
enormous propagation distances from source to detector,
not the small differences in the source behavior. As a result,
we will use the standard general relativistic wave generation framework for the source waveform.
With this new waveform model, we then carry out a
simplified (angle-averaged) Fisher-matrix analysis to estimate the accuracy to which the parameter ðAÞ could be
constrained as a function of , given a gravitational-wave
detection consistent with general relativity. We perform
this study with a waveform model that represents a nonspinning, quasicircular, compact binary inspiral, but that
deviates from general relativity only through the effect of
the modified dispersion relation on the propagation speed
of the waves, via Eq. (1).
To illustrate our results, we show in Table I the accuracy
to which Lorentz-violation in the ¼ 3 case could be
constrained, as a function of system masses and detectors
for fixed signal-to-noise ratio (SNR). The case ¼ 3 is a
TABLE I. Accuracy to which graviton mass and the Lorentzviolating parameter A could be constrained for the ¼ 3 case,
given a gravitational-wave detection consistent with GR. The
first column lists the masses of the objects considered, the
instrument analyzed and the signal-to-noise ratio (SNR).
Detector
m1
m2
mg ðeVÞ
AðeV1 Þ
Ad. LIGO
SNR ¼ 10
1.4
1.4
10
1.4
10
10
3:71 1022
3:56 1022
3:51 1022
7:36 108
3:54 107
6:83 107
ET
SNR ¼ 50
10
10
100
10
100
100
2:99 1023
4:81 1023
6:67 1023
2:32 108
1:12 106
3:34 106
NGO
SNR ¼ 100
104
104
105
105
106
104
105
105
106
106
3:05 1025
2:46 1025
2:03 1025
2:09 1025
1:49 1025
2:16 102
0.147
0.189
9.57
23.2
prediction of ‘‘doubly special relativity.’’ The bounds on
the graviton mass are consistent with previous studies
[11,14–18] (for a recent summary of current and proposed
bounds on mg see [19]). The table here means that given a
gravitational-wave detection consistent with GR, mg and A
would have to be smaller than the numbers on the third and
fourth columns, respectively.
Let us now compare these bounds with current constraints. The mass of the graviton has been constrained
dynamically to mg 7:6 1020 eV through binary pulsar
observations of the orbital period decay and statically to
4:4 1022 eV with Solar System constraints (see e.g.
[19]). We see then that even with the inclusion of an additional A parameter, the projected gravitational-wave bounds
on mg are still interesting. The quantity A has not been
constrained in the gravitational sector. In the electromagnetic sector, the dispersion relation of photon has been constrained: for example, for ¼ 3, A & 1025 eV1 using
TeV -rayobservations [20]. One should note, however, that
such bounds on the photon dispersion relation are independent of those we study here, as in principle the photon and the
graviton dispersion relations need not be tied together.
We must stress that this paper deals only with Lorentzviolating corrections to the gravitational-wave dispersion
relation, and thus, it deals only with propagation effects
and not with generation effects. Generation effects will in
principle be very important, possible leading to the excitation of additional polarizations, as well as modifications
to the quadrupole expressions. Such is the case in several
modified gravity theories, such as the Einstein-Aether
theory and the Horava-Lifshitz theory [21–33].
Generically studying the generation problem, however, is
difficult as there does not exist a general Lagrangian density that can capture all Lorentz-violating effects. Instead,
one would have the gargantuan task of solving the generation problem within each specific theory.
024041-2
CONSTRAINING LORENTZ-VIOLATING, MODIFIED . . .
PHYSICAL REVIEW D 85, 024041 (2012)
The goal of this paper, instead, is to consider generic
Lorentz-violating effects in the dispersion relation and
focus only on the propagation of gravitational waves.
This will then allow us to find the corresponding ppE
parameters that represent Lorentz-violating propagation.
Thus, if future gravitational-wave observations peak at
these ppE parameters, then one could suspect that some
sort of Lorentz-violation could be responsible for such
deviations from General Relativistic. Future work will
concentrate on the generation problem.
The remainder of this paper deals with the details of the
calculations and is organized as follows. In Sec. II,
we introduce and motivate the modified dispersion relation
(1), and derive from it the gravitational-wave speed as a
function of energy and the new Lorentz-violating parameters. In Sec. III, we study the propagation of gravitons in a
cosmological background as determined by the modified
dispersion relation and graviton speed. We find the relation
between emission and arrival times of the gravitational
waves, which then allows us in Sec. IV to construct a
restricted post-Newtonian (PN) gravitational waveform
to 3.5 PN order in the phase ½Oðv=cÞ7 . We also discuss
the connection to the ppE framework. In Sec. V, we calculate the Fisher information matrix for Ad. LIGO, ET, and
a LISA-like mission and determine the accuracy to which
the compact binary’s parameters can be measured, including a bound on the graviton and Lorentz-violating
Compton wavelengths. In Sec. VI, we present some conclusions and discuss possible avenues for future research.
II. THE SPEED OF LORENTZ-VIOLATING
GRAVITATIONAL WAVES
In general relativity, gravitational waves travel at the
speed of light c because the gauge boson associated with
gravity, the graviton, is massless. Modified gravity theories, however, predict modifications to the gravitationalwave dispersion relation, which would in turn force the
waves to travel at speeds different than c. The most intuitive, yet purely phenomenological modification one
might expect is to introduce a mass for the graviton,
following the special relativistic relation
E2 ¼ p2 c2 þ m2g c4 :
(2)
From this dispersion relation, together with the definition v=c p=p0 , or v c2 p=E, one finds the graviton
speed [11]
2 4
mg c
v2g
¼1 2 ;
2
E
c
(3)
where mg , vg , and E are, respectively, the graviton’s
rest mass, velocity, and energy.
Different alternative gravity theories may predict different dispersion relations from Eq. (2). A few examples of
such relations include the following:
(i) Double Special Relativity Theory [34–37]: E2 ¼
p2 c2 þ m2g c4 þ dsrt E3 þ . . . , where dsrt is a parameter of the order of the Planck length.
(ii) Extra-Dimensional
Theories
[38]:
E2 ¼
2
2
2
4
4
p c þ mg c edt E , where edt is a constant
related to the square of the Planck length;
(iii) Hořava-Lifshitz Theory [39–42]: E2 ¼ p2 c2 þ
ð4hl 2hl =16Þp4 þ . . . , where hl and hl are constants of the theory;
(iv) Theories with Non-Commutative Geometries [42–
44]: E2 g21 ðEÞ ¼ m2g c4 þ p2 c2 g22 ðEÞ with g2 ¼ 1
pffiffiffiffiffiffiffiffiffiffiffiffiffi
and g1 ¼ ð1 ncg =2Þexpðncg E2 =E2p Þ, with
ncg a constant.
Of course, the list above is just representative of a few
models, but there are many other examples where the
graviton dispersion relation is modified [45,46]. In general,
a modification of the dispersion relation will be accompanied by a change in either the Lorentz group or its action
in real or momentum space. Lorentz-violating effects of
this type are commonly found in quantum-gravitational
theories, including loop quantum gravity [47] and string
theory [48,49].
Modifications to the standard dispersion relation are usually suppressed by the Planck scale, so one might wonder
why one should study them. Recently, Collins, et al. [50,51]
suggested that Lorentz violations in perturbative quantum
field theories could be dramatically enhanced when one
regularizes and renormalizes them. This is because terms
that would vanish upon renormalization due to Lorentz
invariance do not vanish in Lorentz-violating theories, leading to an enhancement after renormalization [52].
Although this is an appealing argument, we prefer here
to adopt a more agnostic viewpoint and simply ask the
following question: What type of modifications would
enter gravitational-wave observables because of a modified
dispersion relation and to what extent can these deviations
be observed or constrained by current and future
gravitational-wave detectors? In view of this, we postulate
the parametrized dispersion relation of Eq. (1).
One can see that this model-independent dispersion
relation can be easily mapped to all the ones described
above, in the limit where E and p are large compared to
mg , but small compared to the Planck energy Ep . More
precisely, we have
(i) Double Special Relativity: A ¼ dsrt and ¼ 3.
(ii) Extra-Dimensional Theories: A ¼ edt and ¼ 4.
(iii) Hořava-Lifshitz: A ¼ 4hl 2hl =16 and ¼ 4, but
with mg ¼ 0.
(iv) Non-Commutative Geometries: A ¼ 2ncg =E2p and
¼ 4, after renormalizing mg and c.
Of course, for different values of ðA; Þ we can parameterize other Lorentz-violating corrections to the dispersion
relation. One might be naively tempted to think that a p3 or
024041-3
MIRSHEKARI, YUNES, AND WILL
PHYSICAL REVIEW D 85, 024041 (2012)
4
p correction to the above dispersion relation will induce a
1.5 or 2PN correction to the phase relative to the massive
graviton term. This, however, would be clearly wrong, as p
is the graviton’s momentum, not the momentum of the
members of a binary system.
With this modified dispersion relation the modified
graviton speed takes the form
m2g c4
v2g
2 v
¼ 1 2 AE
:
(4)
c
E
c2
To first order in A, this can be written as
m2g c4
m2g c4 =2
v2g
¼ 1 2 AE2 1 2
;
E
E
c2
(5)
and in the limit E mg it takes the form
2 4
mg c
v2g
¼ 1 2 AE2 :
2
E
c
(6)
Notice that if A > 0 or if m2g c4 =E2 > jAjE2 , then the
graviton travels slower than light speed. On the other hand,
if A < 0 and m2g c4 =E2 < jAjE2 , then the graviton would
propagate faster than light speed.
III. PROPAGATION OF GRAVITATIONAL WAVES
WITH A MODIFIED DISPERSION RELATION
We now consider the propagation of gravitational waves
that satisfy the modified dispersion relation of Eq. (1).
Consider the Friedman-Robertson-Walker background
ds2 ¼ dt2 þ a2 ðtÞ½d2 þ 2 ðÞðd
2 þ sin2 d2 Þ; (7)
where aðtÞ is the scale factor with units of length, and ðÞ
is equal to , sin, or sinh if the universe is spatially flat,
closed, or open, respectively. Here and henceforth, we use
units with G ¼ c ¼ 1, where a useful conversion factor is
1M ¼ 4:925 106 s ¼ 1:4675 km.
In a cosmological background, we will assume that the
modified dispersion relation takes the form
g p p ¼ m2g Ajpj ;
E
¼
d
;
dt
p ¼ a2 p ;
(9)
we obtain
2 ð1=2Þ
m2g a2
1
a
d
;
¼ 1þ 2 þA
a
p
p
dt
(10)
2
e ¼
Z ta dt
1 m2g Z ta
aðtÞdt
2 a2 ðte ÞE2e te
te aðtÞ
Z ta
1
Aðaðte ÞEe Þ2
aðtÞ1 dt:
2
te
(11)
Consider gravitons emitted at two different times te
and t0e , with energies Ee and E0e , and received at corresponding arrival times (e is the same for both). Assuming
_ then
te te t0e a=a,
D 1
1
ta ¼ ð1 þ ZÞ te þ 02 2 02
2g fe fe
D
1
1
þ 2 2 02 ;
(12)
fe
fe
2A
where Z a0 =aðte Þ 1 is the cosmological redshift, and
where we have defined
A hA1=ð2Þ ;
(13)
and where mg =Ee ¼ ðg fe Þ1 , with fe the emitted
gravitational-wave frequency, Ee ¼ hfe and g ¼ h=mg
the graviton Compton wavelength. Notice that when
¼ 2, then the A correction vanishes. Notice also that
A always has units of length, irrespective of the value
of . The distance measure D is defined by
1 þ Z 1 Z ta
aðtÞ1 dt;
(14)
D a0
te
where a0 ¼ aðta Þ is the present value of the scale factor.
For a dark energy-matter dominated universe D and the
luminosity distance DL have the form
D ¼
(8)
where jpj ðgij pi pj Þ1=2 . Consider a graviton emitted radially at ¼ e and received at ¼ 0. By virtue of the independence of the t part of the metric, the component p of its 4-momentum is constant along its worldline.
Using E ¼ p0 , together with Eq. (8) and the relations
p
where
¼ a ðte ÞðE2e m2g Ajpje Þ. The overall minus
sign in the above equation is included because the graviton
travels from the source to the observer.
Expanding to first order in ðmg =Ee Þ 1, and
A=p2 1 and integrating from emission time
( ¼ e ) to arrival time ( ¼ 0), we find
p2
ð1 þ ZÞ1 Z Z
ð1 þ z0 Þ2 dz0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
H0
0
M ð1 þ z0 Þ3 þ (15)
1þZ ZZ
dz0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
H0
0
M ð1 þ z0 Þ3 þ (16)
DL ¼
where H0 72 km s1 Mpc1 is the value of the Hubble
parameter today and M ¼ 0:3 and ¼ 0:7 are the
matter and dark energy density parameters, respectively.
Before proceeding, let us comment on the time shift
found above in Eq. (12). First, notice that this equation
agrees with the results of [11] in the limit A ! 0.
Moreover, in the limit ! 0, our results map to those of
2
2
[11] with the relation 2
g ! g þ A . Second, notice
that in the limit ! 2, the ðaðte ÞEe Þ2 in Eq. (11) goes to
024041-4
CONSTRAINING LORENTZ-VIOLATING, MODIFIED . . .
PHYSICAL REVIEW D 85, 024041 (2012)
unity and the A correction becomes frequencyindependent. This makes sense, since in that case the
Lorentz-violating correction we have introduced acts as a
renormalization factor for the speed of light.
IV. MODIFIED WAVEFORM IN THE
STATIONARY-PHASE APPROXIMATION
We consider the gravitational-wave signal generated by
a nonspinning, quasicircular inspiral in the post-Newtonian
approximation. In this scheme, one assumes that orbital
velocities are small compared to the speed of light (v 1)
and gravity is weak (m=r 1). Neglecting any amplitude
corrections (in the so-called restricted PN approximation),
the plus- and cross-polarizations of the metric perturbation
can be represented as
hðtÞ AðtÞeiðtÞ ;
ðtÞ c þ 2
Zt
(17)
fðtÞdt;
(18)
tc
where AðtÞ is an amplitude that depends on the
gravitational-wave polarization (see e.g. Eq. (3.2) in
[11]), while fðtÞ is the observed gravitational-wave frequency, and c and tc are a fiducial phase and fiducial
time, respectively, sometimes called the coalescence phase
and time.
The Fourier transform of Eq. (17) can be obtained
analytically in the stationary-phase approximation, where
we assume that the phase is changing much more rapidly
than the amplitude [53,54]. We then find
~
~ ¼ qAðtÞ
hðfÞ
ffiffiffiffiffiffiffiffi eiðfÞ ;
_
fðtÞ
while for ¼ 1, we find
Z fe
D0 D1
fe
¼1 ðfÞ ¼ 2
ðte tec Þdfe þ
ln
A
fec
fe 2g
fec
c :
(23)
þ 2ftc and 4
Þ are new coalescence
The quantities ðtc ; tc Þ and ð c ; c
times and phases, into which constants of integration have
been absorbed.
We can relate te tec to fe by integrating the frequency
chirp equation for nonspinning, quasicircular inspirals
from general relativity [11]:
dfe
96
743 11
11=3
þ ðmfe Þ2=3
¼
ðMe fe Þ
1
336
4
dte
5M2e
þ 4ðmfe Þ ;
(24)
where we have kept terms up to 1PN order. In the calculations that follow, we actually account for corrections up
to 3.5PN order, although we do not show these higher-order
terms here (they can be found, e.g. in [55]).
After absorbing further constants of integration into
Þ, dropping the bars, and reexpressing every c ; tc ; ðtc ; c
thing in terms of the measured frequency f at the detector
[note that f_ 1=2 ¼ ðdfe =dte Þ1=2 =ð1 þ ZÞ], we obtain
~
iðfÞ ; for 0 < f < f
max
~ ¼ AðfÞe
(25)
hðfÞ
0;
for f > fmax ;
with the definitions
~ Au7=6 ;
AðfÞ
(19)
ðfÞ ¼ 2
Zf
fc
ðt tc Þdf þ 2ftc c (21)
In these equations, Me ¼ 3=5 m is the chirp mass of the
source, where ¼ m1 m2 =ðm1 þ m2 Þ is the symmetric
mass ratio.
We can now substitute Eq. (12) into Eq. (21) to relate the
time at the detector to that at the emitter. Assuming that
1, we find
Z fe
D0
1 ðfÞ ¼ 2
ðte tec Þdfe fe 2g
fec
1
D
c ;
þ 2ftc ð1 Þ fe1 2
4
A
4
1
X
½cn þ ‘n lnðuÞun=3 ;
GR ðfÞ ¼ 2ftc c (20)
:
4
(26)
ðfÞ ¼ GR ðfÞ þ ðfÞ;
where f is the gravitational-wave frequency at the detector
and
~ ¼ 4 Me ðMe fe Þ2=3 ;
AðtÞ
5 a0 ðe Þ
rffiffiffiffiffiffi 2
M
A¼
;
30 DL
þ
3 5=3
u
128
n¼0
(27)
where the
pffiffiffinumerical coefficient ¼ 1 for LIGO and ET,
but ¼ 3=2 for a LISA-like mission (because when one
angle-averages, the resulting geometric factors depend
slightly on the geometry of the detector). The coefficients
ðcn ; ‘n Þ can be read up to n ¼ 7, for example, from
Eq. (3.18) in [55]. In these equations, u Mf is a
dimensionless frequency, while M is the measured chirp
mass, related to the source chirp mass by M ¼
ð1 þ ZÞMe . The frequency fmax represents an upper cutoff
frequency where the PN approximation fails.
The dephasing caused by the propagation effects takes a
slightly different form depending on whether 1 or
¼ 1. In the general 1 case, we find
(22)
024041-5
1 ðfÞ ¼ u1 u1 ;
(28)
MIRSHEKARI, YUNES, AND WILL
PHYSICAL REVIEW D 85, 024041 (2012)
where the parameters and are given by
1 2 D0 M
;
2g ð1 þ ZÞ
2 D
M1
:
ð1 Þ 2
ð1 þ ZÞ1
A
bppE
h~ ppE ðfÞ ¼ A~GR ð1 þ ppE uappE ÞeiGR ðfÞþippE u ;
(29)
(30)
In the special ¼ 1 case, we find
¼1 ðfÞ ¼ u1 þ ¼1 lnðuÞ;
(31)
where remains the same, while
¼1 ¼
D1
;
A
where ðppE ; appE ; ppE ; bppE Þ are ppE, theory parameters.
Notice that in the limit ppE ! 0 or ppE ! 0, the ppE
waveform reduces exactly to the SPA GR waveform.
The proposal is then to match-filter with template families of this type and allow the data to select the best-fit
ppE parameters to determine whether they are consistent
with GR.
We can now map the ppE parameters to those obtained
from a generalized, Lorentz-violating dispersion relation:
ppE ¼ 0
(32)
and we have reabsorbed a factor into the phase of
coalescence.
As before, notice that in the limit A ! 0, Eq. (28)
reduces to the results of [11] for a massive graviton. Also
note that, as before, in the limit ! 0, we can map our
2
2
results to those of [11] with 2
g ! g þ A , i.e. in this
limit, the mass of the graviton and the Lorentz-violating A
term become 100% degenerate. In the limit ! 2,
Eq. (12) becomes frequency-independent, which then implies that its integral, Eq. (21), becomes linear in frequency, which is consistent with the ! 2 limit of
Eq. (28). Such a linear term in the gravitational-wave phase
can be reabsorbed through a redefinition of the time of
coalescence, and thus is not observable. This is consistent
with the observation that the dispersion relation with
¼ 2 is equivalent to the standard massive graviton one
with a renormalization of the speed of light. When ¼ 1,
Eq. (12) leads to a 1=f term, whose integral in Eq. (21)
leads to a lnðfÞ term, as shown in Eq. (23). Finally, notice
that, in comparison with the phasing terms that arise in the
PN approximation to standard general relativity, these
corrections are effectively of ð1 þ 3=2ÞPN order, which
implies that the ¼ 0 term leads to a 1PN correction as in
[11], the ¼ 1 case leads to a 2.5PN correction, the ¼ 3
case leads to a 5.5PN correction and ¼ 4 leads to a 7PN
correction. This suggests that the accuracy to constrain A
will deteriorate very rapidly as increases.
ppE ¼ bppE ¼ 1:
(34)
Quantum-gravity inspired Lorentz-violating theories suggest modified dispersion exponents ¼ 3 or 4, to leading
order in E=mg , which then implies ppE parameters
bppE ¼ 2 and 3. Therefore, if after a gravitational wave
has been detected, a Bayesian analysis with ppE templates
is performed that leads to values of bppE that peak around 2
or 3, this would indicate the possible presence of Lorentz
violation [13]. Notice however that the ¼ 1 case cannot
be recovered by the ppE formalism without generalizing it
to include lnu terms. Such effects are analogous to memory
corrections in PN theory.
At this point, we must spell out an important caveat. The
values of that represent Lorentz violation for quantuminspired theories ( ¼ 3, 4) correspond to very high PN
order effects, i.e. a relative 5.5 or 7PN correction, respectively. Any gravitational-wave test of Lorentz violation
that wishes to constrain such steep momentum dependence
would require a very accurate (high PN order) modeling of
the general relativistic waveform itself. In the next section,
we will employ 3.5PN accurate waveforms, which are the
highest order known, and then ask how well and can be
constrained. Since we are neglecting higher than 3.5PN
order terms in the template waveforms, we are neglecting
also any possible correlations or degeneracies between
these terms and the Lorentz-violating terms. Therefore,
any estimates made in the next section are at best optimistic bounds on how well gravitational-wave measurements
could constrain Lorentz violations.
Connection to the Post-Einsteinian framework
Recently, there has been an effort to develop a framework
suitable for testing for deviations from general relativity
in gravitational-wave data. In analogy with the parametrized post-Newtonian (ppN) framework [10,56–60],
the parametrized post-Einsteinian (ppE) framework
[12,13,61] suggests that we deform the gravitational-wave
observable away from our GR expectations in a wellmotivated, parametrized fashion. In terms of the Fourier
transform of the waveform observable in the SPA, the
simplest ppE meta-waveform is
(33)
V. CONSTRAINING A MODIFIED GRAVITON
DISPERSION RELATION
In this section, we perform a simplified Fisher analysis,
following the method outlined for compact binary inspiral
in [62–64], to get a sense of the bounds one could place on
ðg ; A Þ given a gravitational-wave detection that is consistent with general relativity. We begin by summarizing
some of the basic ideas behind a Fisher analysis, introducing some notation. We then apply this analysis to an Adv.
LIGO detector, an ET detector, and a LISA-like mission.
024041-6
CONSTRAINING LORENTZ-VIOLATING, MODIFIED . . .
PHYSICAL REVIEW D 85, 024041 (2012)
A. General considerations
Given a noise power spectrum, Sn ðfÞ, we can define the
inner product of signals h1 and h2 as
ðh1 jh2 Þ 2
Z 1 h~ h~2 þ h~ h~1
1
2
Sn ðfÞ
0
df;
(35)
where h~1 and h~2 are the Fourier transforms of signals 1 and
2, respectively, and star superscript stands for complex
conjugation. The SNR for a given signal h is simply
½h ¼ ðhjhÞ1=2 :
(36)
If the signal depends on a set of parameters a that we wish
to estimate via matched filtering, then the root-meansquare error on parameter a in the limit of large SNR is
(no summation over a implied here)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
a hð
a h
a iÞ2 i ¼ aa :
(37)
The quantity aa is the ða; aÞ component of the variancecovariance matrix, which is the inverse of the Fisher information matrix, ab , defined as
@h @h
ab j
:
(38)
@
a @
b
The off-diagonal elements of the variance-covariance matrix give the parameter correlation coefficients, which we
define as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cab ab = aa bb :
(39)
We will work with an angle-averaged response function,
so that the templates depend only on the parameters:
a ¼ ðlnA; c ; f0 tc ; lnM; ln; ; Þ;
(41)
where we have defined the integrals
IðqÞ Z 1 xq=3
0
gðxÞ
;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D0 M g >
;
ð1 þ ZÞ 1=4
(40)
where each component of the vector a is dimensionless.
We recall that A is an overall amplitude that contains
information about the gravitational-wave polarization
and the beam-pattern function angles. The quantities c
and tc are the phase and time of coalescence, where f0 is a
frequency characteristic of the detector, typically a ‘‘knee’’
frequency, or a frequency at which Sn ðfÞ is a minimum.
The parameters M and are the chirp mass and symmetric mass ratio, which characterize the compact binary
system under consideration. The parameters ð; Þ describe the massive graviton and Lorentz-violating terms,
respectively.
The SNR for the templates in Eq. (25) is simply
¼ 2AðMÞ7=6 f02=3 Ið7Þ1=2 S1=2
;
0
with x f=f0 . The quantity gðxÞ is the rescaled power
spectral density, defined via gðxÞ Sh ðfÞ=S0 for the
detector in question, and S0 is an overall constant. When
computing the Fisher matrix, we will replace the amplitude
A in favor of the SNR, using Eq. (41). This will then lead
to bounds on ð; Þ that depend on the SNR and on a
rescaled version of the moments JðqÞ IðqÞ=Ið7Þ.
In the next subsections, we will carry out the integrals in
Eq. (42), but we will approximate the limits of integration
by certain xmin and xmax [15]. The maximum frequency will
be chosen to be the smaller of a certain instrumental
maximum threshold frequency and that associated with a
gravitational wave emitted by a particle in an innermoststable circular orbit (ISCO) around a Schwarzschild black
hole (BH): fmax ¼ 63=2 1 3=5 M1 . The maximum instrumental frequency will be chosen to be (105 , 103 , 1) Hz
for Ad. LIGO, ET, and LISA-like, respectively. The minimum frequency will be chosen to be the larger of a certain
instrumental minimum threshold frequency and, in the
case of a space mission, the frequency associated with a
gravitational wave emitted by a test-particle 1 yr prior to
reaching the ISCO. The minimum instrumental frequency
will be chosen to be (10, 1, 105 ) Hz for Ad. LIGO, ET,
and a LISA-like mission, respectively.
Once the Fisher matrix has been calculated, we will
invert it using a Cholesky decomposition to find the
variance-covariance matrix, the diagonal components of
which give us a measure of the accuracy to which parameters could be constrained. Let us then define the upper
bound we could place on ð; Þ as 1=2 = and
are numbers. Combining
1=2 =, where and these definitions with Eqs. (29) and (30), we find, for
1, the bounds:
(42)
2
<
A
1=2 M1
j1 j ;
2 D ð1 þ ZÞ1
(43)
(44)
Notice that the direction of the bound on A itself depends
on whether > 2 or < 2; but because A ¼ ðA =hÞ2 ,
all cases yield an upper bound on A. For the case ¼ 1,
we find
A¼1 >
D1
:
1=2
(45)
In the remaining subsections, we set ¼ 0 and ¼ 0
in all partial derivatives when computing the Fisher
matrix, since we derive the error in estimating and about the nominal or a priori general relativity values,
ð; Þ ¼ ð0; 0Þ.
024041-7
MIRSHEKARI, YUNES, AND WILL
PHYSICAL REVIEW D 85, 024041 (2012)
B. Detector spectral noise densities
We model the Ad. LIGO spectral noise density via [65]
8
2 þ0:0764x4
2
;
Sh ðfÞ < 10164ðxf0 7:9Þ þ 2:4 1062 x50 þ 0:08x4:69 þ 123:35 10:23x
1þ0:17x2
¼
:
S0
1;
8
Sh ðfÞ < ½a1 xb1 þ a2 xb2 þ a3 xb3 þ a4 xb4 2 ; f fs
¼
: 1;
S0
f < fs ;
45
Sgal
h ðfÞ ¼ 2:1 10
Sexgal
ðfÞ ¼ 4:2 1047
h
(47)
and
where f0 ¼ 100 Hz, S0 ¼ 1050 Hz1 , fs ¼ 1 Hz, and
a3 ¼ 1:76;
a2 ¼ 0:349;
a4 ¼ 0:409;
b1 ¼ 15:64;
b3 ¼ 0:12;
b4 ¼ 1:10:
The classic LISA design had an approximate spectral
noise density curve that could be modeled via (see, eg.
[15,66]):
1
Sh ðfÞ ¼ minfSNSA
ðfÞ= expðTmission
dN=dfÞ; SNSA
ðfÞ
h
h
gal
ex-gal
þ S ðfÞg þ S
ðfÞ:
(49)
h
h
where
10
-40
10
10
f < fs ;
f
1 Hz
f
1 Hz
7=3
7=3
Hz1 ;
(51)
Hz1 ;
(52)
11=3
dN
3
1 1 Hz
¼ 2 10 Hz
;
df
f
(53)
1
with f ¼ Tmission
the bin size of the discretely Fourier
transformed data for a classic LISA mission lasting a time
Tmission and ’ 4:5 the average number of frequency bins
that are lost when each galactic binary is fitted out.
Recently, the designs of LISA and ET have changed
somewhat. The new spectral noise density curves can be
computed numerically [67–69] and are plotted in Fig. 1.
Notice that the bucket of the NGO noise curve has shifted
to higher frequency, while the new ET noise curve is more
optimistic than the classic one at lower frequencies. The
spikes in the latter are due to physical resonances, but these
will not affect the analysis. In the remainder of this paper,
we will use the new ET and NGO noise curves to estimate
parameters.
(48)
b2 ¼ 2:145;
(46)
f 4
52
SNSA
ðfÞ
¼
9:18
10
þ 1:59 1041
h
1 Hz
f 2
þ 9:18 1038
(50)
Hz1 :
1 Hz
Here, f0 ¼ 215 Hz, S0 ¼ 1049 Hz1 , and fs ¼ 10 Hz is
a low-frequency cutoff below which Sh ðfÞ can be considered infinite for all practical purposes.
The initial ET design postulated the spectral noise
density [65]
a1 ¼ 2:39 1027 ;
f fs ;
10
-36
-42
10
-37
-44
10
-38
New LISA
Classic LISA
Sn(f)
Sn(f)
New ET
Classic ET
10-46
10
10-39
-48
10-50
100
10
101
10
2
103
10
4
-40
10-41
f (Hz)
10-3
10
-2
10
-1
f (Hz)
FIG. 1 (color online). ET (left) and LISA (right) spectral noise density curves for the classic design (dotted) and the new NGO design
(solid).
024041-8
CONSTRAINING LORENTZ-VIOLATING, MODIFIED . . .
106
10
5
10
PHYSICAL REVIEW D 85, 024041 (2012)
106
1.4 M +1.4 M
10 M +1.4 M
10 M +10 M
10
5
4
10
4
10
3
10
3
10
2
10
2
106
10 M +10 M
10 M +100 M
100 M +100 M
10
5
104
104 M
104 M
105 M
5
106 M
10 M
+1045 M
+105 M
+10 M
+106 M
6
+10 M
103
102
101
101
101
0
10
AdLIGO
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
New LISA
New ET
10
0
0
0.5
1
1.5
2.5
2
3
3.5
4
10-1
0
0.5
1
1.5
2
2.5
3
3.5
4
FIG. 2 (color online). Bounds on parameter for different values of , using AdLIGO and ¼ 10 (left panel), ET and ¼ 50
(center panel), and NGO and ¼ 100 (right panel). Vertical lines at ¼ ð0; 2Þ show where the correction becomes 100% degenerate
with other parameters. Each panel contains several curves that show the bound for systems with different masses.
C. Results
We plot the bounds that can be placed on in Fig. 2 as a
function of the parameter. The left panel corresponds
to the bounds placed with Ad. LIGO and ¼ 10
(DL 160 Mpc, Z 0:036 for a double neutron-star inspiral), the middle panel corresponds to ET and ¼ 50
(DL 2000 Mpc, Z 0:39 for a double 10M BH inspiral), and the right panel corresponds to NGO and ¼ 100
(DL 20 000 Mpc, Z 2:5 for a double 105 M BH inspiral). When ¼ 0 or ¼ 2, cannot be measured at all,
as it becomes 100% correlated with either standard massive
graviton parameters. Thus we have drawn vertical lines in
those cases. As the figure clearly shows, the accuracy to
which can be measured deteriorates rapidly as becomes
larger. In fact, once > 4, we find that cannot be confidently constrained anymore because the Fisher matrix
becomes noninvertible (its condition number exceeds 1016 ).
Attempting to constrain values of > 5=3 becomes
problematic not just from a data analysis point of view,
but also from a fundamental one. The PN templates that we
have constructed contain general relativity phase terms up
to 3.5PN order. Such terms scale as u2=3 , which corresponds to ¼ 5=3. Therefore, trying to measure values
of 5=3 without including the corresponding 4PN and
higher-PN order terms is not well-justified. We have done
so here, neglecting any correlations between these higherorder PN terms and the Lorentz-violating terms, in order to
get a rough sense of how well Lorentz-violating modifications could be constrained.
The bounds on and are converted into a lower bound
on g and upper bound on A in Table II for ¼ 3 and
binary systems with different component masses. Given a
gravitational-wave detection consistent with general relativity, this table says that g and A would have to be larger and
smaller than the numbers in the seventh and eight columns of
the table, respectively. In addition, this table also shows the
accuracy to which standard binary parameters could be
measured, such as the time of coalescence, the chirp mass,
and the symmetric mass ratio, as well as the correlation
coefficients between parameters. Different clusters of numbers correspond to constraints with Ad. LIGO (top), New ET
(middle), and NGO (bottom—see caption for further details).
TABLE II. Root-mean-squared errors for source parameters, the corresponding bounds on g and A , and the correlation
coefficients, for the case ¼ 3 and for systems with different masses in units of M . The top cluster uses the Ad. LIGO Sn ðfÞ,
¼ 10, g is in units of 1012 km, A is in units of 1016 km and tc is in msecs. The middle cluster uses the ET Sn ðfÞ, ¼ 50, g is
in units of 1013 km, A is in units of 1015 km and tc is in msecs. The bottom cluster uses a NGO Sn ðfÞ, ¼ 100, g is in units of
1015 km, A is in units of 1010 km and tc is in secs.
Detector
m1
m2
c
tc
M=M
=
g
A
Ad. LIGO
1.4
1.4
10
1.4
10
10
3.61
3.34
4.16
1.80
9.99
31.0
0.0374%
0.267%
2.40%
6.80%
12.8%
72.2%
3.34
3.48
3.53
0.911
4.36
8.40
ET
10 10
10 100
100 100
0.528 1.59
1.12
44.5
5.23
203
0.0174%
0.259%
4.03%
1.70%
6.67%
67.6%
4.15
2.58
1.86
0.0286 0:952 0:986 0.988 0:742 0.875 0.813
1.38
0:974 0:993 0.993 0:872 0.951 0.915
4.12
0:983 0:995 0.996 0:914 0.969 0.947
NGO
104
104
105
105
106
0.264
0.264
0.295
0.351
0.415
1.05 0.00124% 0.368%
5.42 0.00434% 0.383%
9.54 0.0163% 1.33%
142 0.0574% 2.03%
228
0.138%
5.33%
4.06
5.04
6.12
5.93
8.30
0.266
1.81
2.33
118
286
104
105
105
106
106
024041-9
cM
cM
c
cM
c
c
0:962 0:991 0.989 0:685 0.803 0.740
0:977 0:993 0.917 0:830 0.923 0.875
0:978 0:994 0.995 0:874 0.947 0.915
0:957
0:955
0:944
0:961
0:956
0:990
0:991
0:983
0:990
0:986
0.986
0.984
0.986
0.989
0.990
0:636
0:757
0:749
0:938
0:820
0.761
0.884
0.891
0.942
0.935
0.687
0.809
0.823
0.891
0.885
MIRSHEKARI, YUNES, AND WILL
4e+12
PHYSICAL REVIEW D 85, 024041 (2012)
2.8 M
11.4 M
20 M
2.8 M
11.4 M
20 M
3.8e+12
10-15
(km)
A
3.4e+12
g
(km)
3.6e+12
3.2e+12
3e+12
10-16
2.8e+12
0
0.05
0.1
0.15
0.2
0
0.25
0.05
0.1
0.15
0.2
0.25
FIG. 3 (color online). Bounds on g (left) and A (right) as a function of for different total masses, Ad. LIGO, ¼ 10, and ¼ 3.
Although Fig. 2 suggests bounds on of Oð103 –105 Þ for
the ¼ 3 case, the dimensional bounds in Table II suggest
a strong constraint on A . This is because in converting
from to A one must divide by the D3 distance measure.
This distance is comparable to (but smaller than) the
luminosity distance, and thus, the longer the graviton
propagates the more sensitive the constraints are to possible Lorentz violations. Second, notice that the accuracy
to which many parameters can be determined, e.g.
ðtc ; M; Þ, degrades with total mass because the number of observed gravitational-wave cycles decreases.
Third, notice that the bound on the graviton Compton
wavelength is not greatly affected by the inclusion of
an additional parameter in the ¼ 3 case, and is
comparable to the one obtained in [11] for LIGO. In
fact, we have checked that in the absence of A we
recover Table II in [11].
We now consider how these bounds behave as a function
of the mass ratio. Figure 3 plots the bound on the graviton
Compton wavelength (left) and the Lorentz-violating
Compton wavelength A (right) as a function of for Ad.
LIGO and ¼ 3, with systems of different total mass.
Notice that, in general, both bounds improve for comparable
mass systems, even though the SNR is kept fixed.
With all of this information at hand, it seems likely that
gravitational-wave detection would provide useful information about Lorentz-violating graviton propagation. For
example, if a Bayesian analysis were carried out, once a
gravitational wave is detected, and the ppE parameters
peaked around bppE ¼ 2 or 3, this could possibly indicate
the presence of some degree of Lorentz violation.
Complementarily, if no deviation from general relativity
is observed, then one could constrain the magnitude of A
to interesting levels, considering that no bounds exist to
date.
VI. CONCLUSIONS AND DISCUSSION
We studied whether Lorentz symmetry-breaking in the
propagation of gravitational waves could be measured with
gravitational waves from nonspinning, compact binary
inspirals. We considered modifications to a massive graviton dispersion relation that scale as Ap , where p is the
graviton’s momentum while ðA; Þ are phenomenological
parameters. We found that such a modification introduces
new terms in the gravitational-wave phase due to a delay in
the propagation: waves emitted at low frequency, early in
the inspiral, travel slightly slower than those emitted at
high frequency later. This results in an offset in the relative
arrival times at a detector, and thus, a frequency-dependent
phase correction. We mapped these new gravitationalwave phase terms to the recently proposed ppE scheme,
with ppE phase parameters bppE ¼ 1.
We then carried out a simple Fisher analysis to get a
sense of the accuracy to which such dispersion relation
deviations could be measured with different gravitationalwave detectors. We found that indeed, both the mass of the
graviton and additional dispersion relation deviations
could be constrained. For values of > 4, there is not
enough information in the waveform to produce an invertible Fisher matrix. Certain values of , like ¼ 0 and 2,
also cannot be measured, as they become 100% correlated
with other system parameters.
In deriving these bounds, we have made several approximations that force us to consider them only as rough
indicators that gravitational waves can be used to constrain
generic Lorentz-violation in gravitational-wave propagation. For example, we have not accounted for precession or
eccentricity in the orbits, the merger phase of the inspiral,
the spins of the compact objects or carried out a Bayesian
analysis. We expect the inclusion of these effects to modify
and possibly worsen the bounds presented above by
roughly an order of magnitude, based on previous results
for bounds on the mass of the graviton [11,14,16,70–72].
However, pthe
ffiffiffiffi detection of N gravitational waves would
lead to a N improvement in the bounds [19], while the
modeling of only the Lorentz-violating term, without including the mass of the graviton, would also increase the
accuracy to which A could me measured [13].
024041-10
CONSTRAINING LORENTZ-VIOLATING, MODIFIED . . .
Future work could concentrate on carrying out a more
detailed data analysis study, using Bayesian techniques. In
particular, it would be interesting to compute the evidence
for a general relativity model and a modified dispersion
relation model, given a signal consistent with general
relativity, to see the betting-odds of the signal favoring
GR over the non-GR model. A similar study was already
carried out in [13], but there a single ppE parameter was
considered. Another interesting avenue for future research
would be to consider whether there are any theories
(quantum-inspired or not) that predict fractional powers
or values of different from 3 or 4.
ACKNOWLEDGMENTS
We would like to thank Leo Stein for help streamlining
our Mathematica code. S. M. acknowledges Francesc
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